ABM: Brownian motion, Brownian bridge, geometric Brownian motion,...

In Sim.DiffProc: Simulation of Diffusion Processes

Description

The (S3) generic function for simulation of brownian motion, brownian bridge, geometric brownian motion, and arithmetic brownian motion.

Usage

BM(N, ...)
BB(N, ...)
GBM(N, ...)
ABM(N, ...)

## Default S3 method:
BM(N =1000,M=1,x0=0,t0=0,T=1,Dt=NULL, ...)
## Default S3 method:
BB(N =1000,M=1,x0=0,y=0,t0=0,T=1,Dt=NULL, ...)
## Default S3 method:
GBM(N =1000,M=1,x0=1,t0=0,T=1,Dt=NULL,theta=1,sigma=1, ...)
## Default S3 method:
ABM(N =1000,M=1,x0=0,t0=0,T=1,Dt=NULL,theta=1,sigma=1, ...)

Arguments

N	number of simulation steps.
M	number of trajectories.
x0	initial value of the process at time \code{t0}.
y	terminal value of the process at time \code{T} of the BB.
t0	initial time.
T	final time.
Dt	time step of the simulation (discretization). If it is NULL a default Dt = (T-t0)/N.
theta	the interest rate of the ABM and GBM.
sigma	the volatility of the ABM and GBM.
...	potentially further arguments for (non-default) methods.

Details

The function BM returns a trajectory of the standard Brownian motion (Wiener process) in the time interval [t0,T]. Indeed, for W(dt) it holds true that W(dt) = W(dt) - W(0) -> N(0,dt) -> sqrt(dt) * N(0,1), where N(0,1) is normal distribution Normal.

The function BB returns a trajectory of the Brownian bridge starting at x0 at time t0 and ending at y at time T; i.e., the diffusion process solution of stochastic differential equation:

dX(t) = ((y-X(t))/(T-t)) dt + dW(t)

The function GBM returns a trajectory of the geometric Brownian motion starting at x0 at time t0; i.e., the diffusion process solution of stochastic differential equation:

dX(t) = theta X(t) dt + sigma X(t) dW(t)

The function ABM returns a trajectory of the arithmetic Brownian motion starting at x0 at time t0; i.e.,; the diffusion process solution of stochastic differential equation:

dX(t) = theta dt + sigma dW(t)

Value

X	an visible ts object.

Author(s)

A.C. Guidoum, K. Boukhetala.

References

Allen, E. (2007). Modeling with Ito stochastic differential equations. Springer-Verlag, New York.

Jedrzejewski, F. (2009). Modeles aleatoires et physique probabiliste. Springer-Verlag, New York.

Henderson, D and Plaschko, P. (2006). Stochastic differential equations in science and engineering. World Scientific.

See Also

This functions BM, BBridge and GBM are available in other packages such as "sde".

Examples

op <- par(mfrow = c(2, 2))

## Brownian motion
set.seed(1234)
X <- BM(M = 100)
plot(X,plot.type="single")
lines(as.vector(time(X)),rowMeans(X),col="red")

## Brownian bridge
set.seed(1234)
X <- BB(M =100)
plot(X,plot.type="single")
lines(as.vector(time(X)),rowMeans(X),col="red")

## Geometric Brownian motion
set.seed(1234)
X <- GBM(M = 100)
plot(X,plot.type="single")
lines(as.vector(time(X)),rowMeans(X),col="red")

## Arithmetic Brownian motion
set.seed(1234)
X <- ABM(M = 100)
plot(X,plot.type="single")
lines(as.vector(time(X)),rowMeans(X),col="red")

par(op)

Sim.DiffProc documentation built on Nov. 8, 2020, 4:27 p.m.